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SageMath
E = EllipticCurve("br1")
E.isogeny_class()
Elliptic curves in class 47040br
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
47040.ch4 | 47040br1 | \([0, -1, 0, -144125, -21011955]\) | \(2748251600896/2205\) | \(265642030080\) | \([2]\) | \(196608\) | \(1.4976\) | \(\Gamma_0(N)\)-optimal |
47040.ch3 | 47040br2 | \([0, -1, 0, -145105, -20710703]\) | \(175293437776/4862025\) | \(9371850821222400\) | \([2, 2]\) | \(393216\) | \(1.8441\) | |
47040.ch5 | 47040br3 | \([0, -1, 0, 31295, -68021183]\) | \(439608956/259416045\) | \(-2000161228600442880\) | \([2]\) | \(786432\) | \(2.1907\) | |
47040.ch2 | 47040br4 | \([0, -1, 0, -337185, 45864225]\) | \(549871953124/200930625\) | \(1549224319426560000\) | \([2, 2]\) | \(786432\) | \(2.1907\) | |
47040.ch6 | 47040br5 | \([0, -1, 0, 1034815, 323282625]\) | \(7947184069438/7533176175\) | \(-116165265825801830400\) | \([2]\) | \(1572864\) | \(2.5373\) | |
47040.ch1 | 47040br6 | \([0, -1, 0, -4782465, 4026167937]\) | \(784478485879202/221484375\) | \(3415397529600000000\) | \([4]\) | \(1572864\) | \(2.5373\) |
Rank
sage: E.rank()
The elliptic curves in class 47040br have rank \(1\).
Complex multiplication
The elliptic curves in class 47040br do not have complex multiplication.Modular form 47040.2.a.br
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.